# Overview

This section of the website contains various tools that I have built for helping students learn different concepts from the physical sciences. I'll update this as I develop new tools, but I anticipate that most will be GUIs (graphical user interfaces) for developing intuition for different mathematical concepts. For now, I am developing the interactive tools in Wolfram's computable document format (.cdf). The CDF Player that allows you to view the CDF files is free and available for download here: https://www.wolfram.com/player/

# Translating and Scaling a Parabola

In order to work with this GUI, please download the translating and scaling a parabola CDF file: transAndScaleParabola.cdf

This GUI shows two parabolas, a blue one: $f(x) = x^2$, and a red one: $f(x) = s(x - t)^2$. The parameter s is the scaling constant and the parameter t is the translating constant. Note how changing these parameters affects the red function. Try adjusting the parameters so that the red function exactly matches the blue one. What values for $s$ and $t$ make the two functions equivalent? Plug these values into the red equation and simplify to demonstrate to yourself that these values make the two functions equivalent.

# Exponential Functions

The function in this CDF shows an exponential function with a coefficient $b$ in the exponent, which can be manipulated by the user to hold some value between -2 and 2, and a multiplicative scaling constant $a$ that can also be adjusted by the user. Note that negative values of $b$ demonstrate exponential decay, while positive values show exponential growth.

# Squared Exponential Functions

As you adjust the slider to change the value of $\mu$ for the squared exponential function, make a note of how differently the function behaves when $\mu$ is negative compared to when it is positive. Keep what you learned in mind as you move on to the Gaussian function below.

# The Gaussian Function

The Gaussian function is a statistical distribution named after Carl Friedrich Gauss and is also known as a normal distribution. It is widely used in chemistry and physics. Open up the gaussDemo CDF file and try moving the sliders that control the values of $\mu$ (the mean) and σ (the standard deviation). Have you noticed that the Gaussian function looks very much like a squared exponential function with a negative scaling in the exponent (e.g. $e^{-x^2}$)? Have you also noticed that the mean parameter $\mu$ translates the Gaussian distribution the same way the translating constant did in the parabola example? Another very important thing to note is the fact that the entire function is scaled by a constant ($1 / \sigma \sqrt{2 \pi}$). Thanks to this scaling, the integral of the Gaussian function over all values of x is equal to 1 no matter what values of $\sigma$ and $\mu$ we choose, a necessary quality for a statistical distribution, called being normalized.

# Finite Difference Derivatives

If we have a function $f(x)$ and its derivative, $f'(x)$, the value returned by $f'(x)$ is the slope of $f(x)$ at $x$. That slope has the same slope as a tangent to the curve of $f(x)$ at the point $x$. If you open this CDF file and set $\Delta x$ to the smallest possible value then move the slider for $x$ (the red dot), you will see a visualization where a tangent line to $f(x)$ is drawn wherever you place the red dot. What you are actually seeing, however, are two overlapping lines, where one is the true tangent obtained using the derivative to assign the slope (the red line), and the other is a linear approximation whereby a linear equation is fit to $x$ and a second point $x + \Delta x$ (the cyan dot). The approximation is quite good when the change is small compared to how wiggly the line is. Another way to think about it is that if we zoom into any smooth curve far enough, that tiny piece of the curve will look like a straight line, so treating it as a straight line within that window is a very good approximation.

# Integrals and Numerical Integration

In this CDF learning tool, we have a polynomial $f(x) = .5x^3 + 2x^2+x-1$ that we'd like to integrate, and where the free parameters are both the limits of integration (xMin and xMax) and the number of boxes that we want to use for taking the numerical integral. Calculating the integral of a function outputs the area under that function (where areas below the x-axis contribute negatively to the area). So just as the true value of a derivative at a given point can be thought of as the slope at that position as we take an infinitessimally small step forward (see finite difference CDF above), we can think of taking the integral as adding up the area of a bunch of rectangles under the curve that are infinitessimally thin in width. Play with the sliders a bit and prove to yourself that the numerical approximation to the integral gets closer to the true value when you use many thin boxes rather than just a few thicker ones. It's a bit like measuring in centimeters vs. meters, in that centimeters give you more precision.